A006899 -id:A006899 - cp's OEIS frontend

A334151 Numbers k such that k / rad(k) > m / rad(m) for all m < k.

1, 4, 8, 16, 27, 32, 64, 128, 243, 256, 512, 1024, 2048, 4096, 6561, 8192, 16384, 32768, 59049, 65536, 131072, 262144, 524288, 1048576, 1594323, 2097152, 4194304, 8388608, 14348907, 16777216, 33554432, 67108864, 129140163, 134217728, 268435456

Offset: 1

Ilya Gutkovskiy, Apr 16 2020

nonn

The terms listed in the Data section are numbers of the form 2^i or 3^ceiling(j*(1 + sqrt(2))), i >= 2, j >= 0 (empirical observation).

Michael De Vlieger, Table of n, a(n) for n = 1..4190
Michael De Vlieger, 256 X 256 pixel bitmap showing black if 3 | a(n) else white for n = 1..2^16, read in rows from left to right, stacked top to bottom.

Cf. A003557, A006899, A007947, A064549.

pp = 4; nn = 2^29; j = 0; c = e[_] = 1; r = Prime@ Range[pp];
Do[(e[#1]++; Set[{k, m}, {#1^#2, #1^(#2 - 1)}]) & @@
  First@ MinimalBy[Array[{#, e[#]} &[r[[#]]] &, pp], Power @@ # &];
 If[m > j, Set[{a[c], j}, {k, m}]; c++];
 If[k > nn/2, Break[]], {n, Infinity}];
{1}~Join~Array[a, c - 2, 2] (* Michael De Vlieger, Mar 11 2023 *)

A342621 Sum of the partition number of the prime factors of n with multiplicity.

Original entry on oeis.org

0, 2, 3, 4, 7, 5, 15, 6, 6, 9, 56, 7, 101, 17, 10, 8, 297, 8, 490, 11, 18, 58, 1255, 9, 14, 103, 9, 19, 4565, 12, 6842, 10, 59, 299, 22, 10, 21637, 492, 104, 13, 44583, 20, 63261, 60, 13, 1257, 124754, 11, 30, 16, 300, 105, 329931, 11, 63, 21, 493, 4567, 831820

Offset: 1

Eric Desbiaux, Mar 16 2021

nonn

			For n = 408 = 2^3*3*17, a(408) = 3 * A000041(2) + A000041(3) + A000041(17) = 3*2 + 3 + 297 = 306.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 3000 terms from Eric Desbiaux)

Cf. A000041, A027746, A058698, A001222.

a:= n-> add(combinat[numbpart](i[1])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 17 2021

{0}~Join~Array[Total@ Map[#2 PartitionsP[#1] & @@ # &, FactorInteger[#]] &, 58, 2] (* Michael De Vlieger, Mar 17 2021 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]*numbpart(f[k,1])); \\ Michel Marcus, Mar 17 2021

def a(n):
    return sum([Partitions(primefactor).cardinality() for (primefactor,exponent) in factor(n) for _ in range(exponent)])
[a(n) for n in (1..100)]

a(A003586(n)) - A001414(A003586(n)) = 0.
a(A006899(n)) * A008480(A006899(n)) - A001414(A006899(n)) = 0.
a(n) = Sum_{k=1..A001222(n)} A000041(A027746(n,k)). - Alois P. Heinz, Apr 09 2021

A364902 Let x, y be the greatest exponents of 2, 3 respectively such that 2^x, 3^y do not exceed n and let k_2, k_3 be n - 2^x, and n - 3^y respectively. Then for n such that k_2 = 0 or k_3 = 0, a(n) = n, else a(n) is the least novel number Min{pa(k_2), qa(k_3)}, where p, q are primes not equal to either 2 or 3.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 15, 8, 9, 7, 14, 20, 25, 35, 50, 16, 11, 22, 21, 28, 55, 70, 75, 40, 45, 49, 27, 13, 26, 33, 44, 32, 17, 34, 39, 52, 65, 98, 100, 56, 63, 77, 80, 121, 110, 105, 140, 112, 143, 154, 147, 196, 245, 135, 91, 130, 165, 220, 160, 85, 170, 195, 260, 64, 19, 38, 51, 68

Offset: 1

David James Sycamore and Michael De Vlieger Sep 21 2023

nonn

Motivated by the recursion D(2) known to reproduce A005940, this sequence uses a compound version based on a squarefree semiprime (6) rather than a prime, in which the terms are generated by a greedy algorithm related to the distances between n and the greatest powers of 2, and 3 not exceeding n. After a(9) = 9 each power of 2 or 3 is followed by the smallest prime not yet in the sequence. (e.g. 11 follows 16, 13 follows 27, etc).
There are no multiples of 6 in this sequence.
For k > 2, if a(i) = prime(k) = p and a(j) = p^2 then j-i is a term in A006899 (e.g. a(17) = 11, a(44) = 121 and 44 - 17 = 27 = 3^3).
Conjectures: (i). This is a permutation of A047253 with primes in order; (ii). All terms between consecutive prime terms, prime(k), prime(k+1) are prime(k)-smooth.

			a(n) = n for n <= 4 because all such n are powers of 2 or 3.
a(5) = least novel Min{a(1)*p,a(2)*q} = Min{p,2*q} for o,q prime != 2 or 3, so a(5) = 5.
17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.
Data can be shown in tabular form in two distinct ways: First row starts with 1 and then rows start with a prime; alternatively each row starts with 2^i or 3^j:
 1;                       1;
 2;                       2;
 3,4;                     3;
 5,10,15,8,9;             4,5,10,15;
 7,14,20,25,35,50,16;     8;
 11,22,21,28,55...        9,7,14,20,25,35,50

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A005940, A006899, A047253, A108906, A356867, A364611, A364628.

nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;
Array[Set[{q[#1], p[#1],
      r[#1]}, {#1, #2,
        Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,
       Prime[#2]}] & @@ {#, s[[#]]} &, w];
Do[If[n == 1,
   Set[{a[n], c[1]}, {1, True}],
   Array[Set[m[#], 1] &, w];
   Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];
   Array[
    If[j[#] == 0,
      k[#] = n; flag = #,
      While[Set[k[#], Prime[m[#]] a[j[#]]];
       Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];
   If[flag > 0,
    Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,
    Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 24 2023 *)

For n > 6, a(A006899(n) + 1) = prime(n-2).

A237514 Numbers k such that 2^(k-1) < 3^(m-1) < 2^k < 3^m < 2^(k+1), for some m > 2, a(1) = 1.

Original entry on oeis.org

1, 4, 7, 12, 15, 20, 23, 26, 31, 34, 39, 42, 45, 50, 53, 58, 61, 64, 69, 72, 77, 80, 85, 88, 91, 96, 99, 104, 107, 110, 115, 118, 123, 126, 129, 134, 137, 142, 145, 148, 153, 156, 161, 164, 169, 172, 175, 180, 183, 188, 191, 194, 199, 202, 207, 210, 213, 218, 221, 226, 229, 232, 237, 240

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Feb 08 2014

nonn

Exponents of A006899(n) such that A006899(n-1) and A006899(n+1) are both odd.
Probably finite? The last term?
Subsequence of primes starts 7, 23, 31, 53, 61, 107, 137, 191, 199, 229,...
Prime indices of A006899(n) such that A006899(n-1) and A006899(n+1) are both odd: 2, 7, 43, 113, 131, 139, 149, 157, 193, 211, 263, 281, 307, 317, 379,...
Let f(n) := floor( n * log(2) / log(3)), then k is in the sequence if and only if k = 1 or f(k - 1) = f(k) - 1 and f(k + 1) = f(k) + 1. - Michael Somos, Feb 24 2014

			a(2) = 4 because k = 4 and 2^(4-1) < 3^(3-1) < 2^4 < 3^3 < 2^(4+1) for m = 3;
a(3) = 7 because k = 7 and 2^(7-1) < 3^(4-1) < 2^7 < 3^4 < 2^(7+1) for m = 4;
a(4) = 12 because k = 12 and 2^(12-1) < 3^(8-1) < 2^12 < 3^8 < 2^(12+1) for m = 8.

Cf. A006899 (numbers of the form 2^i or 3^j).

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A334151 Numbers k such that k / rad(k) > m / rad(m) for all m < k.

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A342621 Sum of the partition number of the prime factors of n with multiplicity.

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A237514 Numbers k such that 2^(k-1) < 3^(m-1) < 2^k < 3^m < 2^(k+1), for some m > 2, a(1) = 1.

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